Everything about Elvis Presley was big. Graceland, his iconic Memphis estate, attracts 600,000 people a year to tour the King’s lavish mansion and gawk at his jumpsuits, gold records and memorabilia.

Now the company that manages Graceland is making it even bigger, with a $125 million expansion that includes a new hotel and entertainment complex. But some people aren’t thrilled by one part of the project—the $79 million in local and state tax breaks through which taxpayers are helping to pay for it.

The Graceland incentives are the latest in a string of tax breaks handed out by Memphis, Tenn. to spur economic development, and they have made the city a flash point in a broader debate: Are the $70 billion that state and local governments award in such deals each year a smart way to boost economies, or a giveaway to private businesses which share too few of the risks?

More cities may join the debate: A new rule the Governmental Accounting Standards Board approved in August will soon require state and local governments to disclose how much tax revenues are reduced each year by “abatements” they grant. That could put pressure on government officials to rein in or justify the breaks.

Memphis officials expect the Graceland project to create hundreds of jobs and boost the city’s struggling economy. They say the tax incentives were critical to making the expansion a reality, and that ultimately the city will make back more than what it is laying out.

But some of the incentives irk people like Mike Williams, president of the Memphis Police Association and a candidate in the city’s mayoral election Thursday. He has called for a freeze on new local tax-incentive projects.

“We need a pause button,” said Mr. Williams, citing the more than 400 police officers who have left their jobs recently due to frozen wages and reduced benefits that result from city spending cuts even as Memphis foregoes tax revenues to help Graceland and other businesses. “We need to hold these companies accountable.”

The Graceland expansion centers on the Guest House at Graceland, a new hotel adjacent to the mansion, with 450 rooms and 16,000 square feet of meeting space. Already under construction, it will open in October 2016, replacing Graceland’s smaller, aging Heartbreak Hotel. A planned 200,000-square-foot entertainment complex would house Elvis memorabilia, shops and restaurants.

Memphis and the state of Tennessee are helping pay for the project in three ways. Some of the sales taxes collected at Graceland, half of future increases in Graceland’s property taxes and a new 5% surcharge on merchandise sold at the site will all be steered toward paying off the project’s upfront financing, which was led by J.P. Morgan Chase & Co.’s Highbridge investment arm.

Together, the three tax breaks have a present value of about $78.6 million over 30 years, said James McLaren, an attorney for Elvis Presley Enterprises Inc., which manages Graceland’s operations and other Elvis-related tourism and entertainment ventures.

Memphis has granted tax incentives in the past to companies such as Electrolux AB, International Paper Co. and IKEA, to try to spur job creation and economic development. The Memphis metropolitan area has an unemployment rate of 6.4%, among the highest of any large U.S. metro area. Those abatements meant Shelby County, which includes Memphis, didn’t collect $48.7 million in property taxes in 2014 that it otherwise would have been entitled to.

Some question whether Memphis has handed out its incentives too freely, and on terms not favorable to the city. Electrolux received $188 million from Memphis in tax breaks and other incentives to build a kitchen-oven manufacturing plant, but the deal doesn’t allow the city to “claw back” the incentives if Electrolux fails to create the 1,240 jobs it promised.

As of 2014, Electrolux’s plant had created 968 jobs, according to a performance report it filed with the city, and it still has until 2018 to meet its target. An Electrolux spokeswoman said the company was “ahead of our job growth commitment.”

Reid Dulberger, chief economic development officer for Memphis and Shelby County, said Electrolux insisted on certain ground rules among the cities that competed for the plant, including “limited incentive compliance.” But he was “confident” the company would meet its targets, and he said the city’s overall track record on tax incentives is sound.

According to an August report by Memphis Consulting Group LLC, a local consulting firm, 12 out of 47 Memphis property tax-break projects active as of 2013 had fallen short on commitments for either jobs, wages or capital investment. City officials say some of those projects had their tax breaks reduced.

Some think the Graceland incentives are particularly egregious because Graceland doesn’t have the leverage to threaten to move elsewhere, as other businesses seeking tax breaks often do.

“Elvis Presley Enterprises shouldn’t have much in the way of bargaining power at all, unless they want to shut down entirely,” said Kenneth Thomas, a University of Missouri-St. Louis professor who has criticized Memphis over the Graceland incentives.

Extending tax incentives hasn’t always worked out well for state and local governments. In Atlantic City, N.J., the Revel casino was approved for up to $261.3 million in state tax reimbursements, but never turned a profit and later filed for bankruptcy. A spokeswoman for New Jersey’s economic-development agency said Revel had “not received one penny” of that award.

In Albuquerque, N.M., a unit of Germany’s Schott AG opened a solar-panel factory with the help of $130 million in tax breaks and other subsidies, but the plant closed in 2012, leaving the state out $16 million. Schott didn’t provide any comment. Jon Barela, New Mexico’s secretary of economic development, said the state has revised its incentive program since the Schott deal, which was reached under a previous administration. He believes tax breaks can be “an effective economic development tool,” however.

Government officials and business executives maintain the tax breaks pay for themselves by encouraging job creation and generating more spending. They also put dormant properties to work and help revive distressed areas. And given the competition among states and cities to attract business, the officials say, they are almost a requirement.

The project will be “generating revenue for the city and county that would not otherwise be there,” said Graceland’s Mr. McLaren.

Specifically, the expansion is projected to create more than 500 new jobs and generate $50 million in new tax revenue over 15 years, over and above the incentive amounts.

The city predicts the Graceland property involved in the expansion, for instance, will generate $1.2 million in city property taxes a year, about four times what it brought to Memphis in 2013. Even if half of the increase goes toward paying off the construction costs, the city will still collect more taxes than before.

Thomas Malone, president of the Memphis Fire Fighters Association, is concerned that the burden of Memphis’s foregone tax revenues falls on the backs of city employees and hurts public services. He doesn’t oppose all tax breaks: “if it works, then it’s great.” But “if it doesn’t, we’re left holding the bag.”